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arxiv: 2606.22739 · v1 · pith:M4XY72ZRnew · submitted 2026-06-22 · 📡 eess.SY · cs.SY· physics.comp-ph

Development, Validation, and Benchmarking of a Multidisciplinary Semi-Analytical Model for Wave Energy Converters

Rebecca McCabe , Madison Dietrich , Maha Haji This is my paper

Pith reviewed 2026-06-26 07:36 UTC · model grok-4.3

classification 📡 eess.SY cs.SYphysics.comp-ph
keywords wave energy converterssemi-analytical modelmultidisciplinary optimizationtechno-economic analysisoptimal controlhydrodynamic solverdesign explorationbenchmarking
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The pith

MDOcean delivers a semi-analytical framework for wave energy converters that runs in 151 milliseconds while agreeing with higher-fidelity tools to within a few percent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents MDOcean as an open-source framework that integrates hydrodynamics, dynamics, structures, and economics through analytical and semi-analytical methods for early-stage WEC design. It employs an eigenfunction-based hydrodynamic solver, a quasi-linearized dynamics engine, a structural sizing module with multiple failure criteria, and a linearized pseudo-spectral optimal control formulation to treat nonlinearities and constraints. Validation shows the 151 ms runtime is orders of magnitude faster than leading simulation tools with agreement within a few percent in most cases. A sympathetic reader would care because existing tools cannot support the large-scale parametric studies and multidisciplinary optimization needed to balance power, cost, and survivability.

Core claim

MDOcean integrates an eigenfunction-based linear hydrodynamic solver, a quasi-linearized frequency-domain dynamics engine capable of modeling drag and saturation nonlinearities, a structural sizing module incorporating yield, ultimate, buckling, storm, and fatigue criteria, and a simple cost model, all unified through a linearized pseudo-spectral optimal control formulation that extends frequency-domain constraint-handling with describing functions and an analytical quadratically-constrained quadratic program. Validation and benchmarking demonstrate that this architecture achieves a 151 ms runtime while maintaining agreement with higher-fidelity baselines to within a few percent in most case

What carries the argument

The linearized pseudo-spectral optimal control formulation that unifies describing-function treatment of nonlinearities with an analytical quadratically-constrained quadratic program to preserve frequency-domain compatibility and optimization tractability.

If this is right

  • Enables parametric analysis and multidisciplinary optimization over design spaces that are computationally prohibitive for existing tools.
  • Supports integrated techno-economic studies that simultaneously consider power production, structural integrity, and cost.
  • Reveals scaling laws and subsystem interactions that govern overall WEC performance.
  • Provides rapid screening capability before committing resources to higher-fidelity verification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reported speed opens the possibility of evaluating thousands of candidate designs in the time previously required for a single high-fidelity run.
  • Open release of the framework may encourage community additions of further nonlinear effects or refined cost models without altering the core speed advantage.
  • Designers could adopt the tool for initial down-selection, reserving slower numerical codes only for the final shortlist of candidates.

Load-bearing premise

The eigenfunction-based solver, quasi-linearized dynamics, and pseudo-spectral control capture enough of the essential physics and economics that agreement with higher-fidelity baselines within a few percent validates the framework for early-stage design.

What would settle it

A direct head-to-head comparison on a specific WEC geometry and sea state where MDOcean's predicted average power or peak structural load deviates by more than a few percent from experimental data or a verified high-fidelity time-domain simulation.

Figures

Figures reproduced from arXiv: 2606.22739 by Madison Dietrich, Maha Haji, Rebecca McCabe.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simplified XDSM diagram Optimal Control Technique Explicit Implicit Simultaneous Nested MDF Analysis Technique SAND Optimizer Solver Upstream Modules Downstream Modules Dynamics + Control Module State Residual Design Coeffs Power J, g Optimizer Upstream Modules Downstream Modules Dynamics + Control Module Residual Design, state Coeffs Power J, g State Design Optimizer Upstream Modules Downstream Modules Dy… view at source ↗
Figure 3
Figure 3. Figure 3: Optimization architectures organized by analysis method (MDF and SAND) and control method (explicit, implicit, simultaneous, and nested). requirements) and non-monotonic, motivating concurrent rather than sequential optimization. MDO Architecture All modules except dynamics are explicit. The dynamics module requires an internal fixed￾point iteration to resolve nonlinearities in the quasi-linear frequency-d… view at source ↗
Figure 4
Figure 4. Figure 4: Dimension labeling of system Δzs = h - Ts h Δzf = h – Tf,2 H 2π/k _ [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Venn diagram comparing different methods for control synthesis, with MDOcean’s quasi-linearized pseudo-spectral approach at the intersection of frequency domain, nonlinear, and constrained optimal control. See text for acronym descriptions. R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 21 of 75 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: MEEM geometry. Dashed lines represent geometry that is neglected in the MEEM calculations Frequency-domain synthesis Constraint-aware optimal control Nonlinear controller synthesis Imped. matching Loop shaping PID tuning H∞, H2 Linear MPC Constr. LQR Trajectory opt. Feedback lin. Sliding mode Backstepping Lyapunov Latching/declutching Harmonic bal. Descr. fn. Spec. lin. Constr. loop shpg. Lin. spectral OC … view at source ↗
Figure 9
Figure 9. Figure 9: Conceptual demonstration of describing function approximations. Plots show force versus time. Nonlinear signals in red; fundamental amplitudes in blue. R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 22 of 75 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Power matrix multiplication [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Applied loads and fixed points of each structure R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 23 of 75 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Error breakdown based on WEC-Sim Verification Runs Module Time Breakdown Geometry Hydrodynamics Dynamics (2-DOF) Structures Economics 0 0.05 0.1 0.15 0.2 Runtime (s) 0.003 0.236 0.078 0.021 0.001 [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Bar chart showing simulation runtime breakdown between modules Hydrodynamics Runtime Comparison for a single frequency 10 -2 10 -1 10 0 MDOcean MEEM Capytaine Runtime (s) 0.011 0.323 [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Bar chart demonstrating the speed improvement of MDOcean’s hydro module over baseline solver Capytaine Dynamics Runtime Comparison for all sea states Freq. Domain, Unconstrained MDOcean, Unconstrained MDOcean, Force Constrained MDOcean, Fully Constrained WecSim (Parallel), Unconstrained 10 -2 10 0 10 2 Runtime (s) 0.017 0.052 0.07 0.078 291 [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: Effect of wave environment and hydrodynamic design variables on (a) radiation efficiency and (b) capture width ratio 10 -1 10 0 Radiation E/ciency CW=CWmax 10 -6 10 -4 10 -2 10 0 10 2 10 4 S u r fa c e A r e a / Wav ele n g t h 2 0.01 1.505 3 a2=h 1.01 2.505 4 a3=a1 a 1 /a2 d 1 /h d 2 /d1 [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Normalized effect of damping plate aspect ratio on maximum stress and deflection. The dashed lines indicate the nominal design point at 𝑏∕𝑎 = 0.2. R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 25 of 75 [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Effect of Force Limit on Annual Average Power and Peak Structural Load PTO Sweep for Nominal RM3 Geometry Avg Power (kW) 5 10 F max (MN) 100 200 300 400 500 600 700 800 900 1000 P max (kW) 35 40 45 50 55 60 65 70 75 Design Cost ($M) 5 10 F max (MN) 100 200 300 400 500 600 700 800 900 1000 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 LCOE ($/kWh) 5 10 F max (MN) 100 200 300 400 500 600 700 800 900 1000 0.9 1 1.1 1.2 1.3 … view at source ↗
Figure 21
Figure 21. Figure 21: Design of experiments R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 26 of 75 [PITH_FULL_IMAGE:figures/full_fig_p028_21.png] view at source ↗
Figure 20
Figure 20. Figure 20: Effect of Generator Force Limit and Power Limit on Annual Average Power and LCOE Design of Experiments Results 0.6 0.7 0.8 0.9 1 1.1 LCOE ($/kWh) D s Df T f,2 h s h fs,clear F max P max t fb t sr t d h stiff,f h 1,stiff,d 0.5 1 1.5 2 2.5 3 Design Variable Ratio (-) 1.8 2 2.2 2.4 2.6 2.8 3 Structural & PTO Cost ($M) [PITH_FULL_IMAGE:figures/full_fig_p028_20.png] view at source ↗
Figure 22
Figure 22. Figure 22: Power matrix decomposition R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 27 of 75 [PITH_FULL_IMAGE:figures/full_fig_p029_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Fluid regions and dimensions used in the dual concentric cylinder MEEM 0 5 10 15 20 0 2 4 6 8 10 12 14 16 18 20 [PITH_FULL_IMAGE:figures/full_fig_p054_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: A-matrix sparsity pattern, shown for 𝑁 = 𝑀 = 𝐾 = 4 R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 52 of 75 [PITH_FULL_IMAGE:figures/full_fig_p054_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Nondimensional added mass and damping coefficient validation against (Chau and Yeung, 2012) -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 Z 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 Magnitude of Potential Potential Matching Potential 1 at a1 Potential 2 at a1 Potential 2 at a2 Potential e at a2 Chau & Yeung 2012 at a1 Chau & Yeung 2012 at a2 [PITH_FULL_IMAGE:figures/full_fig_p055_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Matching for 𝑁 = 𝑀 = 𝐾 = 11 for benchmark geometry R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 53 of 75 [PITH_FULL_IMAGE:figures/full_fig_p055_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Convergence for 𝑁 = 𝑀 = 𝐾 = (5, 10, 20, 30) for RM3 10 -3 10 -2 10 -1 10 0 10 1 m 0 h/acosh(realmax) (-) -10 -50 -10 -100 -10 -150 -10 -200 -10 -250 -10 -300 b N+2M+1 (m 2 ) b Asymptotic b 1/2 1/(d 2 /h) [PITH_FULL_IMAGE:figures/full_fig_p056_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Asymptotic b-vector for large 𝑚0ℎ R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 54 of 75 [PITH_FULL_IMAGE:figures/full_fig_p056_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Nondimensionalized hydrodynamic data used to interpolate spar coefficients, showing the rebound effect. ˆζ γf ˆζ γs ˆζ Zi −Fˆ i,f+ +Fˆ i,s− + Fˆ p,s − + Fˆ p,f − + Fˆ e,f − + Fˆ e,s − ˆ˙ξf → ˆ˙ξs → Intrinsic PTO kinematics hkin PTO dynamics, load not shown + FˆP T O − X ˆ˙ P T O → = Fˆ th Zi,th + FˆP T O − X ˆ˙ P T O → Th´evenin equivalent intrinsic and PTO kinematics PTO dynamics, load not shown [PITH_F… view at source ↗
Figure 30
Figure 30. Figure 30: Multiport circuit with powertrain kinematics represented as a hybrid matrix, followed by the Thévenin equivalent of the intrinsic dynamics and kinematics R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 55 of 75 [PITH_FULL_IMAGE:figures/full_fig_p057_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: PTO dynamics represented as a multiport circuit and electrical Thévenin equivalent -5 0 5 -2 0 2 4 6 8 10 Optimal occurs at intersection Active constraints Inactive constraints Origin Candidate points Optimal point -5 0 5 -4 -2 0 2 4 6 8 Infeasible -5 0 5 -6 -4 -2 0 2 4 6 8 Optimal occurs at origin -5 0 5 -2 0 2 4 6 8 Optimal occurs at closest point [PITH_FULL_IMAGE:figures/full_fig_p058_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Visualization of the optimization problem Equation (78) in the complex plane of Γ. Toy problems with various constraint circle centers Γ𝑐,𝜇, all with radius 𝑟𝜇 = 4 and 𝑆𝜇 = 1, demonstrate the various solution cases in Equation (80). R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 56 of 75 [PITH_FULL_IMAGE:figures/full_fig_p058_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Nondimensional drag integrals shown in polar coordinates for various values of 𝜅, where the polar radius represents 1∕𝑟 and the polar angle represents 𝜃. 9max;slam H=2 Infeasible if "zslam H=2 < 1 and j3 ! :j < :=2 0 0.2 0.4 0.6 0.8 1 3=: 0 0.5 1 1.5 2 2.5 3 " z sla m H = 2 -0.5 0 0.5 1 1.5 2 2.5 9min;slam H=2 Unde-ned if "zslam H=2 < jsin 3j 0 0.2 0.4 0.6 0.8 1 3=: 0 0.5 1 1.5 2 2.5 3 " z sla m H = 2 -2.… view at source ↗
Figure 34
Figure 34. Figure 34: Nondimensional critical slamming amplitude R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 57 of 75 [PITH_FULL_IMAGE:figures/full_fig_p059_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Slamming Model Results for Nominal RM3 Float Design in Operational Sea States R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 58 of 75 [PITH_FULL_IMAGE:figures/full_fig_p060_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: Dynamic validation showing error in unsaturated mechanical power generation for a variety of modeling assumptions 59 [PITH_FULL_IMAGE:figures/full_fig_p061_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: Drag force comparison showing large errors in drag force, despite low errors in amplitude and power 61 [PITH_FULL_IMAGE:figures/full_fig_p063_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: Total Harmonic Distortion of Float Displacement in WEC-Sim Simulations with Drag Float Drag Force: WEC-Sim vs Describing Function Approximation 0 5 Time (s) -4000 -2000 0 2000 4000 Fo r c e (N) H = 0:5 m, T = 6 s WEC-Sim Drag Force cos(!t + ?)j cos(!t + ?)j approx 0 5 10 Time (s) -4000 -2000 0 2000 4000 Fo r c e (N) H = 0:5 m, T = 12 s 0 5 10 Time (s) -400000 -200000 0 200000 400000 Fo r c e (N) H = 4:8 m… view at source ↗
Figure 39
Figure 39. Figure 39: Comparison of Assumed and Actual Drag Force Signal Shape R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 68 of 75 [PITH_FULL_IMAGE:figures/full_fig_p070_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: Comparison of mechanical power for RM3 report Neary et al. (2014b) and MDOcean 10 0 10 1 10 2 N WEC 0 #10 0 5 #10 7 1 #10 8 1.5 #10 8 2 #10 8 2.5 #10 8 3 #10 8 3.5 #10 8 4 #10 8 Capex ($) 10 0 10 1 10 2 N WEC 1 #10 6 2 #10 6 3 #10 6 4 #10 6 5 #10 6 6 #10 6 7 #10 6 8 #10 6 9 #10 6 1 #10 7 Opex ($) 10 0 10 1 10 2 N WEC 0.5 1 1.5 2 2.5 3 3.5 4 4.5 LCOE ($/kWh) 10 0 10 1 10 2 N WEC 1.6 #10 6 1.8 #10 6 2 #10 6… view at source ↗
Figure 41
Figure 41. Figure 41: Validation for cost scaling with number of WECs teq t zmax Neutral axis [PITH_FULL_IMAGE:figures/full_fig_p071_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: Equivalent thickness for a stiffened plate R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 69 of 75 [PITH_FULL_IMAGE:figures/full_fig_p071_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: Trapezoidal float plate with inscribed circle used to determine 𝑥min Equivalent section and flexural moduli (AISI B5) Concentrated load on annular plate (Boedo and Prantil) Distributed load on annular plate (Roark’s table 11.2 case 2L) Tubular support stiffness 𝐾௧௨௕௘ = 𝐹௧௨௕௘/𝛿௧௨௕௘ 𝛿̅௣௟௔௧௘,௖௢௡ = 𝛿௣௟௔௧௘,௖௢௡𝐷௘௤ 𝐹௧௨௕௘𝑎ଶ 𝑀ഥ௥,௖௢௡ = 𝑀௥,௖௢௡/𝐹௧௨௕௘ Solve for 𝐹௧௨௕௘ with compatibility: 𝛿௣௟௔௧௘ = 𝛿௧௨௕௘ 𝑆௘௤ 𝐷௘௤ 𝛿̅௣௟௔௧௘,… view at source ↗
Figure 44
Figure 44. Figure 44: Calculation process flowchart for the damping plate structural assessment R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 70 of 75 [PITH_FULL_IMAGE:figures/full_fig_p072_44.png] view at source ↗
Figure 45
Figure 45. Figure 45: Damping plate structural analysis: radial bending moment (a) and deflection (b) over the plate radius. R. McCabe et al.: Preprint submitted to Applied Ocean Research Page 71 of 75 [PITH_FULL_IMAGE:figures/full_fig_p073_45.png] view at source ↗
read the original abstract

Wave energy converters (WECs) require system-level techno-economic analysis to balance power production, cost, and survivability. Existing simulation tools are either too computationally costly for large-scale optimization or too narrow in disciplinary scope to support integrated design studies. This work presents MDOcean, a novel open-source WEC simulation framework for rapid early-stage design exploration, parametric analysis, and multidisciplinary optimization. MDOcean integrates hydrodynamics, dynamics, structures, and economics in a computationally efficient architecture based on analytical and semi-analytical methods that substantially reduce runtime while maintaining near-numerical accuracy. The framework includes an eigenfunction-based linear hydrodynamic solver, a quasi-linearized frequency-domain dynamics engine capable of modeling drag and saturation nonlinearities, a structural sizing module incorporating realistic yield, ultimate, buckling, storm, and fatigue design criteria, and a simple cost model for techno-economic assessment. Particular emphasis is placed on the linearized pseudo-spectral optimal control formulation, which extends frequency-domain constraint-handling approaches with a unified describing-function and analytical quadratically-constrained quadratic program framework. This formulation efficiently treats nonlinearities and constraints while preserving compatibility with optimization and frequency-domain analysis techniques. Validation and benchmarking demonstrate that MDOcean's 151 ms runtime is orders of magnitude faster than leading WEC simulation tools while maintaining agreement with higher-fidelity baselines to within a few percent in most cases. The framework also provides insight into limiting behaviors, scaling laws, subsystem interactions, and key tradeoffs governing WEC design and techno-economic performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents MDOcean, an open-source multidisciplinary semi-analytical simulation framework for wave energy converters (WECs) that integrates an eigenfunction-based linear hydrodynamic solver, a quasi-linearized frequency-domain dynamics engine (including drag and saturation nonlinearities), a structural sizing module with yield/ultimate/buckling/storm/fatigue criteria, a simple cost model, and a linearized pseudo-spectral optimal control formulation based on describing functions and QCQP. The central claims are that the framework achieves a runtime of 151 ms (orders of magnitude faster than leading tools) while maintaining agreement with higher-fidelity baselines to within a few percent in most cases, enabling rapid early-stage design, parametric analysis, and techno-economic optimization.

Significance. If the reported accuracy and timing hold under the provided benchmarks, the work supplies a practical, reproducible tool for integrated WEC design studies that were previously limited by computational cost or disciplinary scope. The open-source release, explicit benchmark comparisons against external higher-fidelity codes, and derivation of the semi-analytical components (eigenfunction hydrodynamics, quasi-linearized dynamics, and QCQP control) are notable strengths that support reproducibility and falsifiability of the performance claims.

major comments (2)
  1. [Validation and benchmarking] Validation section: the claim of agreement 'within a few percent in most cases' requires explicit per-case error metrics (e.g., RMS or peak power error, with standard deviations or error bars) and a clear statement of the validation cases, data exclusion rules, and baseline tool versions; without these the quantitative support for the central accuracy claim remains difficult to assess independently.
  2. [Control formulation] § on linearized pseudo-spectral optimal control: the extension from frequency-domain constraint-handling to the unified describing-function + QCQP framework is load-bearing for the nonlinearity treatment claim; the manuscript should include a direct comparison (e.g., Table or Figure) of power capture with and without the describing-function linearization against a time-domain nonlinear reference to confirm the approximation error remains within the stated few-percent band.
minor comments (2)
  1. [Figures] Figure captions for runtime and accuracy plots should explicitly state the hardware platform, number of runs averaged for the 151 ms figure, and the precise definition of 'most cases' for the agreement metric.
  2. [Notation] Notation: the symbols for the describing function and the QCQP matrices should be defined in a single nomenclature table or at first use to avoid ambiguity when reading the control section independently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive review and recommendation for minor revision. We address each major comment below and will incorporate the requested clarifications and comparisons into the revised manuscript.

read point-by-point responses

  1. Referee: [Validation and benchmarking] Validation section: the claim of agreement 'within a few percent in most cases' requires explicit per-case error metrics (e.g., RMS or peak power error, with standard deviations or error bars) and a clear statement of the validation cases, data exclusion rules, and baseline tool versions; without these the quantitative support for the central accuracy claim remains difficult to assess independently.

    Authors: We agree that explicit per-case metrics and additional methodological details are needed to strengthen the quantitative support for the accuracy claim. In the revised manuscript we will add a table reporting RMS and peak power errors (with standard deviations or error bars) for each validation case, together with an explicit enumeration of the cases considered, any data exclusion rules, and the precise versions of the baseline tools employed. revision: yes

  2. Referee: [Control formulation] § on linearized pseudo-spectral optimal control: the extension from frequency-domain constraint-handling to the unified describing-function + QCQP framework is load-bearing for the nonlinearity treatment claim; the manuscript should include a direct comparison (e.g., Table or Figure) of power capture with and without the describing-function linearization against a time-domain nonlinear reference to confirm the approximation error remains within the stated few-percent band.

    Authors: We recognize that a direct head-to-head comparison is required to substantiate the approximation error of the describing-function linearization. We will add a table or figure in the revised manuscript that reports power capture with and without the describing-function linearization, benchmarked against a time-domain nonlinear reference, to verify that the error stays within the few-percent band. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained against external baselines

full rationale

The paper presents a newly developed MDOcean framework with eigenfunction hydrodynamics, quasi-linearized dynamics, structural criteria, and QCQP control. All load-bearing components are formulated from first principles or standard methods and validated directly against independent higher-fidelity external simulation tools, with reported agreement to within a few percent and runtime benchmarks. No equation or claim reduces by construction to a parameter fitted from the authors' own prior outputs, nor does any central result depend on a self-citation chain that itself lacks independent verification. The open-source release further supports external reproduction of the timing and accuracy numbers without reference to internal fits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The model rests on standard domain assumptions in linear hydrodynamics and structural design criteria plus the unverified claim that the semi-analytical approximations preserve accuracy for optimization purposes.

axioms (2)
  • domain assumption Eigenfunction-based linear hydrodynamics and quasi-linearized frequency-domain dynamics sufficiently represent WEC behavior for early-stage design
    Invoked throughout the framework description in the abstract.
  • domain assumption The structural sizing module with yield, ultimate, buckling, storm, and fatigue criteria produces realistic designs
    Stated as part of the integrated modules.

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Reference graph

Works this paper leans on

299 extracted references · 223 canonical work pages · 2 internal anchors

  1. [1]

    Leveraging

    McCabe, Rebecca , month = aug, year =. Leveraging

  2. [2]

    Numerical methods for scientists and engineers , url =

    Hamming, Richard Wesley , year =. Numerical methods for scientists and engineers , url =

  3. [3]

    Speech to the

    Thunberg, Greta , month = apr, year =. Speech to the

  4. [4]

    Matrix structure and convergence behaviour of the matched eigenfunction method for computing heave wave forces on generalized concentric bodies

    Bimali, Yinghui and McCabe, Rebecca and Treacy, Collin and Khanal, Kapil and Lo, En and Haji, Maha , month = may, year =. Matrix structure and convergence behaviour of the matched eigenfunction method for computing heave wave forces on generalized concentric bodies , url =. doi:10.48550/arXiv.2605.19730 , abstract =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.19730

  5. [5]

    Development,

    McCabe, Rebecca and Dietrich, Madison and Haji, Maha , year =. Development,

  6. [6]

    Leveraging

    McCabe, Rebecca and Dietrich, Madison and Haji, Maha , year =. Leveraging

  7. [7]

    and Vandenberghe, Lieven , month = mar, year =

    Boyd, Stephen P. and Vandenberghe, Lieven , month = mar, year =. Convex

  8. [8]

    Optimization Letters , author =

    Disciplined quasiconvex programming , volume =. Optimization Letters , author =. 2020 , keywords =. doi:10.1007/s11590-020-01561-8 , abstract =

    work page doi:10.1007/s11590-020-01561-8 2020

  9. [9]

    IEEE Transactions on Automatic Control , author =

    System analysis via integral quadratic constraints , volume =. IEEE Transactions on Automatic Control , author =. 1997 , keywords =. doi:10.1109/9.587335 , abstract =

    work page doi:10.1109/9.587335 1997

  10. [10]

    IEEE Transactions on Automatic Control , author =

    The pseudospectral. IEEE Transactions on Automatic Control , author =. 1995 , keywords =. doi:10.1109/9.467672 , abstract =

    work page doi:10.1109/9.467672 1995

  11. [11]

    Journal of Offshore Mechanics and Arctic Engineering , author =

    A. Journal of Offshore Mechanics and Arctic Engineering , author =. doi:10.1115/1.4002735 , abstract =

    work page doi:10.1115/1.4002735

  12. [12]

    Applied Ocean Research , author =

    Computationally efficient spectral-domain wave-to-wire modeling of wave energy converters with geared rotary generators , volume =. Applied Ocean Research , author =. 2026 , keywords =. doi:10.1016/j.apor.2026.105028 , abstract =

    work page doi:10.1016/j.apor.2026.105028 2026

  13. [13]

    Nonlinear

    Atherton, Derek P , year =. Nonlinear

  14. [14]

    and Meerkov, Semyon M

    Ching, ShiNung and Eun, Yongsoon and Gokcek, Cevat and Kabamba, Pierre T. and Meerkov, Semyon M. , year =. Quasilinear. doi:10.1017/CBO9780511976476 , abstract =

    work page doi:10.1017/cbo9780511976476

  15. [15]

    , month = dec, year =

    Roberts, John Brian and Spanos, Pol D. , month = dec, year =. Random

  16. [16]

    Journal of Process Control , author =

    A tutorial on linear and bilinear matrix inequalities , volume =. Journal of Process Control , author =. 2000 , pages =. doi:10.1016/S0959-1524(99)00056-6 , abstract =

    work page doi:10.1016/s0959-1524(99)00056-6 2000

  17. [17]

    Ocean Engineering , author =

    Second-order wave excitation forces in. Ocean Engineering , author =. 2026 , keywords =. doi:10.1016/j.oceaneng.2026.124605 , abstract =

    work page doi:10.1016/j.oceaneng.2026.124605 2026

  18. [18]

    McCabe, Rebecca , month = aug, year =

  19. [19]

    Open-source toolbox for semi-analytical hydrodynamic coefficients via the matched eigenfunction expansion method , url =

    McCabe, Rebecca and Khanal, Kapil and Haji, Maha , month = aug, year =. Open-source toolbox for semi-analytical hydrodynamic coefficients via the matched eigenfunction expansion method , url =. doi:10.5281/zenodo.14504017 , abstract =

    work page doi:10.5281/zenodo.14504017

  20. [20]

    Ruehl, Kelley Michelle and Leon-Quiroga, Jorge Andres and Michelen Strofer, Carlos Alejandro and Topper, Mathew and Tom, Nathan and Baca, Elena and Ogden, David , month = may, year =. Next-. doi:10.2172/2431205 , abstract =

    work page doi:10.2172/2431205

  21. [21]

    Best, Hope and Khanal, Kapil and McCabe, Rebecca and Jiang, Ruiyang and Treacy, Collin and Haji, Maha , year =

  22. [22]

    Contraction , isbn =

    Chicone, Carmen , month = may, year =. Contraction , isbn =. Ordinary

  23. [23]

    and Bhattacharyya, S

    Philip, Nimmy Thankom and Nallayarasu, S. and Bhattacharyya, S. K. , year =. Damping. doi:10.1115/OMAE2012-83290 , abstract =

    work page doi:10.1115/omae2012-83290

  24. [24]

    Anderson, Megan and Gaebele, Daniel and Roach, Aeron and Forbrush, Dominic and Roberts, Jesse and Weber, Jochem , month = aug, year =. Re-

  25. [25]

    Ogden, David and Quinton, Zuriah and Lataillade, Tristan de and Pallud, Maxime , year =. 15th. doi:10.36688/ewtec-2023-473 , language =

    work page doi:10.36688/ewtec-2023-473 2023

  26. [26]

    Multidisciplinary

    McCabe, Rebecca and Murphy, Olivia and Haji, Maha , month = nov, year =. Multidisciplinary. doi:10.1115/DETC2022-90227 , abstract =

    work page doi:10.1115/detc2022-90227

  27. [27]

    , year =

    Chau, Fun Pang and Yeung, Ronald W. , year =. Inertia,. doi:10.1115/OMAE2012-83987 , abstract =

    work page doi:10.1115/omae2012-83987

  28. [28]

    Dynamic system design optimization of wave energy converters utilizing direct transcription , copyright =

    Herber, Daniel Ronald , month = may, year =. Dynamic system design optimization of wave energy converters utilizing direct transcription , copyright =

  29. [29]

    Monotonicity-

    Azarm, Shapour and Li, Wei-Chu , year =. Monotonicity-

  30. [30]

    Janzou, Steve , month = nov, year =. System

  31. [31]

    Journal of Dynamic Systems, Measurement, and Control , author =

    Spectral. Journal of Dynamic Systems, Measurement, and Control , author =. doi:10.1115/1.4034780 , abstract =

    work page doi:10.1115/1.4034780

  32. [32]

    Mathematical Programming , author =

    Coordinate descent algorithms , volume =. Mathematical Programming , author =. 2015 , keywords =. doi:10.1007/s10107-015-0892-3 , abstract =

    work page doi:10.1007/s10107-015-0892-3 2015

  33. [33]

    Martins, Joaquim R. R. A. and Lambe, Andrew B. , month = sep, year =. Multidisciplinary. AIAA Journal , publisher =. doi:10.2514/1.J051895 , abstract =

    work page doi:10.2514/1.j051895

  34. [34]

    , month = jul, year =

    Reveyrand, T. , month = jul, year =. Multiport conversions between. 2018 International Workshop on Integrated Nonlinear Microwave and Millimetre-wave Circuits (INMMIC) , publisher =. doi:10.1109/INMMIC.2018.8430023 , abstract =

    work page doi:10.1109/inmmic.2018.8430023 2018

  35. [35]

    Power system analysis , volume =

    Saadat, Hadi , year =. Power system analysis , volume =

  36. [36]

    , month = jan, year =

    Weisstein, Eric W. , month = jan, year =. Cassini. MathWorld - a Wolfram Resource , publisher =

  37. [37]

    Underactuated

    Tedrake, Russ , year =. Underactuated

  38. [38]

    Mechatronics , author =

    Co-design of a wave energy converter through bi-conjugate impedance matching , volume =. Mechatronics , author =. 2025 , keywords =. doi:10.1016/j.mechatronics.2025.103395 , abstract =

    work page doi:10.1016/j.mechatronics.2025.103395 2025

  39. [39]

    and Whitlam, Craig and Chapman, John and Hughes, Jack and Redfearn, Bryony and Brown, Scott and Draper, Scott and Borthwick, Alistair G

    Edwards, Emma C. and Whitlam, Craig and Chapman, John and Hughes, Jack and Redfearn, Bryony and Brown, Scott and Draper, Scott and Borthwick, Alistair G. L. and Foster, Graham and Yue, Dick K.-P. and Hann, Martyn and Greaves, Deborah , month = jun, year =. The effect of device geometry on the performance of a wave energy converter , volume =. Communicatio...

    work page doi:10.1038/s44172-025-00441-2

  40. [40]

    and Patterson, David and Wilson, David , month = apr, year =

    Bacelli, Giorgio and Coe, Ryan G. and Patterson, David and Wilson, David , month = apr, year =. System. Energies , publisher =. doi:10.3390/en10040472 , abstract =

    work page doi:10.3390/en10040472

  41. [41]

    Ocean Engineering , author =

    State-space representation of radiation forces in time-domain vessel models , volume =. Ocean Engineering , author =. 2005 , keywords =. doi:10.1016/j.oceaneng.2005.02.009 , abstract =

    work page doi:10.1016/j.oceaneng.2005.02.009 2005

  42. [42]

    Ocean Engineering , author =

    Added mass and damping of horizontal circular cylinder sections , volume =. Ocean Engineering , author =. 1988 , pages =. doi:10.1016/0029-8018(88)90012-1 , abstract =

    work page doi:10.1016/0029-8018(88)90012-1 1988

  43. [43]

    Dimensional

    McKinley, Garreth and Cheng, W and Brisson, John , year =. Dimensional. 2.006:

  44. [44]

    Idemat scope 3 eco-costs , url =

    van den Herik, Jasper and Vögtlander, Joost , year =. Idemat scope 3 eco-costs , url =

  45. [45]

    Renewable Energy , author =

    Development and validation of a high-resolution regional wave hindcast model for. Renewable Energy , author =. 2020 , keywords =. doi:10.1016/j.renene.2020.01.077 , abstract =

    work page doi:10.1016/j.renene.2020.01.077 2020

  46. [46]

    Renewable Energy , author =

    Development and validation of a regional-scale high-resolution unstructured model for wave energy resource characterization along the. Renewable Energy , author =. 2019 , keywords =. doi:10.1016/j.renene.2019.01.020 , abstract =

    work page doi:10.1016/j.renene.2019.01.020 2019

  47. [47]

    Designing wind turbines for profitability in the day-ahead market , volume =

    Mehta, Mihir Kishore and Zaaijer, Michiel and von Terzi, Dominic , month = dec, year =. Designing wind turbines for profitability in the day-ahead market , volume =. Wind Energy Science , publisher =. doi:10.5194/wes-9-2283-2024 , abstract =

    work page doi:10.5194/wes-9-2283-2024 2024

  48. [48]

    Moraski, Jill and Qvist, Malwina and Spokas, Kasparas , month = may, year =. Beyond

  49. [49]

    Economic

    Makaremi, Elaheh , year =. Economic

  50. [50]

    How do technological choices affect the economic and environmental performance of offshore wind farms? , volume =

    Kainz, S and Guilloré, A and Bottasso, C L , month = jun, year =. How do technological choices affect the economic and environmental performance of offshore wind farms? , volume =. Journal of Physics: Conference Series , publisher =. doi:10.1088/1742-6596/2767/8/082005 , abstract =

    work page doi:10.1088/1742-6596/2767/8/082005

  51. [51]

    , month = jun, year =

    Canet, Helena and Guilloré, Adrien and Bottasso, Carlo L. , month = jun, year =. The eco-conscious wind turbine: design beyond purely economic metrics , volume =. Wind Energy Science , publisher =. doi:10.5194/wes-8-1029-2023 , abstract =

    work page doi:10.5194/wes-8-1029-2023 2023

  52. [52]

    Bonaldo, Luca and Chakrabarti, Sambuddha and Cheng, Fangwei and Ding, Yifu and Jenkins, Jesse D. and Luo, Qian and Macdonald, Ruaridh and Mallapragada, Dharik and Manocha, Aneesha and Mantegna, Gabe and Morris, Jack and Patankar, Neha and Pecci, Filippo and Schwartz, Aaron and Schwartz, Jacob and Schivley, Greg and Sepulveda, Nestor and Xu, Qingyu and Zho...

    work page doi:10.5281/zenodo.14807684

  53. [53]

    and Fripp, Matthias and Bonaldo, Luca , month = mar, year =

    Schivley, Greg and Welty, Ethan and Patankar, Neha and Jacobson, Anna and Xu, Qingyu and Manocha, Aneesha and Pecora, Braden and Bhandarkar, Riti and Jenkins, Jesse D. and Fripp, Matthias and Bonaldo, Luca , month = mar, year =. doi:10.5281/zenodo.15066032 , abstract =

    work page doi:10.5281/zenodo.15066032

  54. [54]

    Ocean Engineering , author =

    Experimental and numerical comparisons of self-reacting point absorber wave energy converters in regular waves , volume =. Ocean Engineering , author =. 2015 , keywords =. doi:10.1016/j.oceaneng.2015.05.027 , abstract =

    work page doi:10.1016/j.oceaneng.2015.05.027 2015

  55. [55]

    Applied Ocean Research , author =

    Maximum wave-power absorption under motion constraints , volume =. Applied Ocean Research , author =. 1981 , pages =. doi:10.1016/0141-1187(81)90063-8 , abstract =

    work page doi:10.1016/0141-1187(81)90063-8 1981

  56. [56]

    and Stehly, T

    Cotrell, J. and Stehly, T. and Johnson, J. and Roberts, J. O. and Parker, Z. and Scott, G. and Heimiller, D. , month = jan, year =. Analysis of. doi:10.2172/1123207 , abstract =

    work page doi:10.2172/1123207

  57. [57]

    and Hadjimichael, Antonia and Malek, Keyvan and Karimi, Tina and Vernon, Chris R

    Reed, Patrick M. and Hadjimichael, Antonia and Malek, Keyvan and Karimi, Tina and Vernon, Chris R. and Srikrishnan, Vivek and Gupta, Rohini S. and Gold, David F. and Lee, Ben and Keller, Klaus and Thurber, Travis B. and Rice, Jennie S. , year =. Addressing. doi:10.5281/zenodo.6110623 , publisher =

    work page doi:10.5281/zenodo.6110623

  58. [58]

    Box, George E. P. and Hunter, William Gordon and Hunter, J. Stuart , year =. Statistics for experimenters: an introduction to design, data analysis, and model building , shorttitle =

  59. [59]

    Hendrikx, R. W. M. and Leth, J. and Andersen, P. and Heemels, W. P. M. H. , month = aug, year =. Optimal control of a wave energy converter , url =. 2017. doi:10.1109/CCTA.2017.8062556 , abstract =

    work page doi:10.1109/ccta.2017.8062556 2017

  60. [60]

    Renewable and Sustainable Energy Reviews , author =

    A practical approach to wave energy modeling and control , volume =. Renewable and Sustainable Energy Reviews , author =. 2021 , keywords =. doi:10.1016/j.rser.2021.110791 , abstract =

    work page doi:10.1016/j.rser.2021.110791 2021

  61. [61]

    , editor =

    Folley, M. , editor =. Spectral-. Numerical. 2016 , keywords =. doi:10.1016/B978-0-12-803210-7.00004-9 , abstract =

    work page doi:10.1016/b978-0-12-803210-7.00004-9 2016

  62. [62]

    Michelén and Topper, Mathew and jtgrasb and Lawson, Michael and Husain, Salman and Shabara, Mohamed and Leon, Jorge and Ling, Bradley A

    Ruehl, Kelley and Keester, Adam and dforbush2 and Ströfer, Carlos A. Michelén and Topper, Mathew and jtgrasb and Lawson, Michael and Husain, Salman and Shabara, Mohamed and Leon, Jorge and Ling, Bradley A. and Ogden, David and j-vanrij and jhbates and Nguyen, Lily and Jeffalo1 and sedwardsand and Davies, Ryan and ratanakso and emiliofa and crobarcro and a...

    work page doi:10.5281/zenodo.14648966

  63. [63]

    Journal of Offshore Mechanics and Arctic Engineering , author =

    Efficient. Journal of Offshore Mechanics and Arctic Engineering , author =. doi:10.1115/1.4032898 , abstract =

    work page doi:10.1115/1.4032898

  64. [64]

    Synergistic design of a combined floating wind turbine - wave energy converter , copyright =

    Kluger, Jocelyn Maxine , year =. Synergistic design of a combined floating wind turbine - wave energy converter , copyright =

  65. [65]

    International Journal of Marine Energy , author =

    Optimal causal control of wave energy converters in stochastic waves –. International Journal of Marine Energy , author =. 2016 , keywords =. doi:10.1016/j.ijome.2016.04.004 , abstract =

    work page doi:10.1016/j.ijome.2016.04.004 2016

  66. [66]

    Applied Energy , author =

    Enhancing the performance of hybrid wave-wind energy systems through a fast and adaptive chaotic multi-objective swarm optimisation method , volume =. Applied Energy , author =. 2024 , keywords =. doi:10.1016/j.apenergy.2024.122955 , abstract =

    work page doi:10.1016/j.apenergy.2024.122955 2024

  67. [67]

    Applied Ocean Research , author =

    Stochastic analysis of the nonlinear dynamics of oscillating water columns:. Applied Ocean Research , author =. 2023 , keywords =. doi:10.1016/j.apor.2023.103711 , abstract =

    work page doi:10.1016/j.apor.2023.103711 2023

  68. [68]

    da Silva, Leandro S. P. and Cazzolato, Benjamin S. and Sergiienko, Nataliia Y. and Ding, Boyin and Morishita, Helio M. and Pesce, Celso P. , month = may, year =. Statistical linearization of the. Journal of Ocean Engineering and Marine Energy , publisher =. doi:10.1007/s40722-020-00165-9 , abstract =

    work page doi:10.1007/s40722-020-00165-9

  69. [69]

    Ocean Engineering , author =

    Slender-body approach for computing second-order wave loads in the frequency domain , volume =. Ocean Engineering , author =. 2025 , keywords =. doi:10.1016/j.oceaneng.2025.120558 , abstract =

    work page doi:10.1016/j.oceaneng.2025.120558 2025

  70. [70]

    Comptes Rendus

    Multiple-gradient descent algorithm (. Comptes Rendus Mathematique , author =. 2012 , pages =. doi:10.1016/j.crma.2012.03.014 , abstract =

    work page doi:10.1016/j.crma.2012.03.014 2012

  71. [71]

    Applied Energy , author =

    Temporal complementarity of marine renewables with wind and solar generation:. Applied Energy , author =. 2022 , keywords =. doi:10.1016/j.apenergy.2022.119276 , abstract =

    work page doi:10.1016/j.apenergy.2022.119276 2022

  72. [72]

    Applied Energy , author =

    Timing value of marine renewable energy resources for potential grid applications , volume =. Applied Energy , author =. 2021 , keywords =. doi:10.1016/j.apenergy.2021.117281 , abstract =

    work page doi:10.1016/j.apenergy.2021.117281 2021

  73. [73]

    Renewable Energy , author =

    From. Renewable Energy , author =. 2025 , keywords =. doi:10.1016/j.renene.2024.122338 , abstract =

    work page doi:10.1016/j.renene.2024.122338 2025

  74. [74]

    Structural and Multidisciplinary Optimization , author =

    Extensions to the design structure matrix for the description of multidisciplinary design, analysis, and optimization processes , volume =. Structural and Multidisciplinary Optimization , author =. 2012 , keywords =. doi:10.1007/s00158-012-0763-y , abstract =

    work page doi:10.1007/s00158-012-0763-y 2012

  75. [75]

    Reference

    Previsic, Mirko , month = sep, year =. Reference

  76. [76]

    A gradient flow approach for combined layout-control design of wave energy parks , url =

    Gambarini, Marco and Ciaramella, Gabriele and Miglio, Edie , month = sep, year =. A gradient flow approach for combined layout-control design of wave energy parks , url =. doi:10.48550/arXiv.2409.10200 , abstract =

    work page doi:10.48550/arxiv.2409.10200

  77. [77]

    and Woinowsky-Krieger, S

    Timoshenko, S. and Woinowsky-Krieger, S. , year =. Theory of

  78. [78]

    , month = sep, year =

    Sobieszczanski-Sobieski, Jaroslaw and Barthelemy, Jean-Francois and Riley, Kathleen M. , month = sep, year =. Sensitivity of. AIAA Journal , publisher =. doi:10.2514/3.51191 , number =

    work page doi:10.2514/3.51191

  79. [79]

    Custódio, A. L. and Madeira, J. F. A. and Vaz, A. I. F. and Vicente, L. N. , month = jul, year =. Direct. SIAM Journal on Optimization , publisher =. doi:10.1137/10079731X , abstract =

    work page doi:10.1137/10079731x

  80. [80]

    Previsic, Mirko , month = sep, year =. 100

Showing first 80 references.